Author's note: This is a continuation of vectors, which is part of the Linear Algebra chapter from Data Science from Scratch by Joel Grus.
Matrices
The first thing to note is that matrices are represented as lists of lists which is explicit with type annotation:
from typing import List
Matrix = List[List[float]]
You might bet wondering if a list of lists is somehow different from a list of vectors we saw previously with the vector_sum function. To see, I used type annotation to try to define the arguments differently.
Here's the vector_sum function we defined previously:
def vector_sum(vectors: List[Vector]) -> Vector:
"""Sum all corresponding elements (componentwise sum)"""
# Check that vectors is not empty
assert vectors, "no vectors provided!"
# Check the vectorss are all the same size
num_elements = len(vectors[0])
assert all(len(v) == num_elements for v in vectors), "different sizes!"
# the i-th element of the result is the sum of every vector[i]
return [sum(vector[i] for vector in vectors)
for i in range(num_elements)]
assert vector_sum([[1,2], [3,4], [5,6], [7,8]]) == [16,20]
Here's a new function, vector_sum2 defined differently with type annotation:
def vector_sum2(lists: List[List[float]]) -> List:
"""Sum all corresponding list (componentwise sum?)"""
assert lists, "this list is empty!"
# check that lists are the same size
num_lists = len(lists[0])
assert all(len(l) == num_lists for l in lists), "different sizes!"
# the i-th list is the sum of every list[i]
return [sum(l[i] for l in lists)
for i in range(num_lists)]
assert vector_sum2([[1,2], [3,4], [5,6], [7,8]]) == [16,20]
I did a variety of things to see if vector_sum and vector_sum2 behaved differently, but they appear to be identical:
# both are functions
assert callable(vector_sum) == True
assert callable(vector_sum2) == True
# when taking the same argument, they both return a list
type(vector_sum([[1,2], [3,4], [5,6], [7,8]])) #list
type(vector_sum2([[1,2], [3,4], [5,6], [7,8]])) #list
# the same input yields the same output
vector_sum([[1,2],[3,4]]) # [4,6]
vector_sum2([[1,2],[3,4]]) # [4,6]
To keep it simple, in the context of matrices, you can think of vectors as the rows of the matrix.
For example, if we represent the small dataset below as a matrix, we can think of columns as variables like: height, weight, age; and each row as a person:
sample_data = [[70, 170, 40],
[65, 120, 26],
[77, 250, 19]]
By extension of rows and columns, we can write a function for the shape of a matrix. This below shape function takes in a matrix and returns a tuple with two integers, number of rows and number of columns:
from typing import Tuple
def shape(A: Matrix) -> Tuple[int, int]:
"""Returns (# of rows of A, # of columns of A)"""
num_rows = len(A)
num_cols = len(A[0]) if A else 0 # number of elements in first row
return num_rows, num_cols
assert shape([[1,2,3], [4,5,6]]) == (2,3) # 2 rows, 3 columns
assert shape(sample_data) == (3,3)
We can actually write functions to grab either a specific row or a specific columns :
Vector = List[float]
# rows
def get_row(A: Matrix, i: int) -> Vector:
"""Returns the i-th row of A (as a Vector)"""
return A[i] # A[i] is already the ith row
# column
def get_column(A: Matrix, i: int) -> Vector:
"""Returns the j-th column of A (as a Vector)"""
return [A_i[j]
for A_i in A]
Now, going beyond finding the shape, rows and columns of an existing matrix, we'll also want to create matrices and we'll do that using nested list comprehensions:
from typing import Callable
def make_matrix(num_rows: int,
num_cols: int,
entry_fn: Callable[[int, int], float]) -> Matrix:
"""
Returns a num_rows x num_cols matrix
whose (i,j)-th entry is entry_fn(i, j)
"""
return [[entry_fn(i,j) # given i, create a list
for j in range(num_cols)] # [entry_fn(i, 0), ...]
for i in range(num_rows)] # create one list for each i
Then we'll actually use the make_matrix function to create a special type of matrix called the identity matrix:
def identity_matrix(n: int) -> Matrix:
"""Returns the n x n identity matrix"""
return make_matrix(n, n, lambda i, j: 1 if i == j else 0)
assert identity_matrix(5) == [[1, 0, 0, 0, 0],
[0, 1, 0, 0, 0],
[0, 0, 1, 0, 0],
[0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]]
Summary
To be sure there are other types of matrices, but in this chapter we're only briefly exploring its construction to prime us.
We know matrices can be used to represent data, each row in the dataset being a vector. Because we can also know a matrices' column, we'll use it to represent linear functions that map k-dimensional vectors to n-dimensional vectors.
Finally, matrices can also be used to map binary relationships.
Flashback to Ch.1
On our first day at DataScienster™ we were given friendship_pairs data:
friendship_pairs = [(0,1), (0,2), (1,2), (1,3), (2,3), (3,4),
(4,5), (5,6), (5,7), (6,8), (7,8), (8,9)]
These friendship_pairs can also be represented in matrix form:
# user 0 1 2 3 4 5 6 7 8 9
friend_matrix = [[0, 1, 1, 0, 0, 0, 0, 0, 0, 0], # user 0
[1, 0, 1, 1, 0, 0, 0, 0, 0, 0], # user 1
[1, 1, 0, 1, 0, 0, 0, 0, 0, 0], # user 2
[0, 1, 1, 0, 1, 0, 0, 0, 0, 0], # user 3
[0, 0, 0, 1, 0, 1, 0, 0, 0, 0], # user 4
[0, 0, 0, 0, 1, 0, 1, 1, 0, 0], # user 5
[0, 0, 0, 0, 0, 1, 0, 0, 1, 0], # user 6
[0, 0, 0, 0, 0, 1, 0, 0, 1, 0], # user 7
[0, 0, 0, 0, 0, 0, 1, 1, 0, 1], # user 8
[0, 0, 0, 0, 0, 0, 0, 0, 1, 0]] # user 9
This allows us to check very quickly whether two users are friends or not:
assert friend_matrix[0][2] == 1, "0 and 2 are friends"
assert friend_matrix[0][8] == 0, "0 and 8 are not friends"
And if we wanted to check for each user's friend, we could:
# checking the friends of user at index five (Clive)
friends_of_five = [i
for i, is_friend in enumerate(friend_matrix[5])
if is_friend]
# checking the friends of user at index zero (Hero)
friends_of_zero = [i
for i, is_friend in enumerate(friend_matrix[0])
if is_friend]
assert friends_of_five == [4,6,7]
assert friends_of_zero == [1,2]
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Top comments (2)
very cool work.
Thank you so much.